What is the value of `{(1-sin^2theta) sec^2theta + tan^2theta} (cos^2theta + 1)`, where `0^@ lt theta lt 90^@`?

Options

A. 2
B. `gt 2`
C. `lt 2`
D. None of the above

Answer

To solve the expression \({(1 - \sin^2 \theta) \sec^2 \theta + \tan^2 \theta} ( \cos^2 \theta + 1)\), we can follow these steps:

### Step 1: Simplify \(1 - \sin^2 \theta\)
Using the Pythagorean identity, we know that:
\[ 1 - \sin^2 \theta = \cos^2 \theta \]
Thus, we can rewrite the expression:
\[ (1 - \sin^2 \theta) \sec^2 \theta = \cos^2 \theta \sec^2 \theta \]

### Step 2: Substitute \(\sec^2 \theta\)
Recall that \(\sec^2 \theta = \frac{1}{\cos^2 \theta}\). Therefore:
\[ \cos^2 \theta \sec^2 \theta = \cos^2 \theta \cdot \frac{1}{\cos^2 \theta} = 1 \]
Now our expression simplifies to:
\[ 1 + \tan^2 \theta \]

### Step 3: Use the identity for \(\tan^2 \theta\)
We know that:
\[ 1 + \tan^2 \theta = \sec^2 \theta \]
Thus, we can rewrite the expression as:
\[ \sec^2 \theta (\cos^2 \theta + 1) \]

### Step 4: Simplify \((\cos^2 \theta + 1)\)
The term \((\cos^2 \theta + 1)\) remains as is. Therefore, our expression now looks like:
\[ \sec^2 \theta (\cos^2 \theta + 1) \]

### Step 5: Evaluate \(\sec^2 \theta (\cos^2 \theta + 1)\)
Now we can expand this:
\[ \sec^2 \theta \cdot \cos^2 \theta + \sec^2 \theta \]
Using \(\sec^2 \theta = 1 + \tan^2 \theta\), we can substitute back:
\[ 1 + \tan^2 \theta \cdot \cos^2 \theta + \sec^2 \theta \]
However, since we already simplified it to \(\sec^2 \theta (\cos^2 \theta + 1)\), we can evaluate it directly.

### Step 6: Conclusion
Since \(\sec^2 \theta\) is always greater than or equal to 1 for \(0 < \theta < 90\), and \(\cos^2 \theta + 1\) is always greater than or equal to 1, the entire expression evaluates to a value greater than 2.

Thus, the final answer is:
\[ \text{The value is greater than 2.} \]